proof of Jensen’s inequality

We prove an equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, more convenient formulation: Let X be some random variableMathworldPlanetmath, and let f(x) be a convex function (defined at least on a segment containing the range of X). Then the expected value of f(X) is at least the value of f at the mean of X:


Indeed, let c=𝔼[X]. Since f(x) is convex, there exists a supporting line for f(x) at c:


for some α, and φ(x)f(x). Then


as claimed.

Title proof of Jensen’s inequality
Canonical name ProofOfJensensInequality
Date of creation 2013-03-22 12:45:15
Last modified on 2013-03-22 12:45:15
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 6
Author Andrea Ambrosio (7332)
Entry type Proof
Classification msc 26D15
Classification msc 39B62